Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains

Communications in Mathematical Physics

Volumn 308, Issue 3, 2011, Pages 641-701

  Kim, Chanwoo ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80955166074     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-011-1355-1     Document Type: Article

References (42)

1. Aoki, K.: Private communications.
2. Arkeryd L., Cercignani C.: A global existence theorem for initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rat. Mech. Anal. 125, 271-287 (1993).
3. Arlotti, L., Banasiak, J., Lods, B.: On general transport equations with abstract boundary conditions. The case of divergence free force field. Preprint 2009.
4. Aoki K., Bardos C., Dogbe C., Golse F.: A note on the propagation of boundary induced discontinuities in kinetic theory. Math. Models Methods Appl. Sci. 11(9), 1581-1595 (2001).
5. Aoki K., Takata S., Aikawa H., Golse F.: A rarefied gas flow caused by a discontinuous wall temperature. Phys. Fluids 13(9), 2645-2661 (2001).
6. Alexandre R., Villani C.: On the Boltzmann equation for long-range interactions. Comm. Pure Appl. Math. 55(1), 30-70 (2002).
7. Boudin L., Desvillettes L.: On the singularities of the global small solutions of the full Boltzmann equation. Monatshefte Math. 131, 91-108 (2000).
8. Bernis L., Desvillettes L.: Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete Contin. Dyn. Syst. 24(1), 13-33 (2009).
9. Cercignani C.: Propagation phenomena in classical and relativistic rarefied gases. Transport Theory Statist. Phys. 29(3-5), 607-614 (2000).
10. Cercignani C.: On the initial-boundary value problem for the Boltzmann equation. Arch. Rat. Mech. Anal. 116, 307-315 (1992).
11. Cercignani C.: The Boltzmann equation and its applications. Springer, New York (1988).
12. Cercignani C., Illner R., Pulvirenti M.: The mathematical theory of dilute gases. Springer, New York (1994).
13. DiPerna R. J., Lions P. L.: On the Cauchy problem for Boltzmann equation: global existence and weak stability. Ann. Math. 130, 321-366 (1989).
14. Duan R., Li M.-R., Yang T.: Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. Math. Models Methods Appl. Sci. 18(7), 1093-1114 (2008).
15. Glassey, R.: The Cauchy Problems in Kinetic Theory. Philadelphia: SIAM, 1996.
16. Greenberg, W., van der Mee, C., Protopopescu, V.: Boundary value problems in abstract kinetic theory. Operator Theory: Advances and Applications, 23. Basel: Birkhauser Verlag, 1987.
17. Gressman T., Strain R.: Global Classical Solutions of the Boltzmann Equation without Angular Cut-off. J. Amer. Math. Soc. 24(3), 771-847 (2011).
18. Guiraud, J.-P.: An H theorem for a gas of rigid spheres in a bounded domain, Theories cinetiques classiques et relativistes. Paris: Centre Nat. Recherche Sci., 1975, pp. 29-58.
19. Guo Y.: Singular solutions of the Vlasov-Maxwell system on a half line. Arch. Rat. Mech. Anal. 131(3), 241-304 (1995).
20. Guo Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Rat. Mech. Anal. 169(4), 305-353 (2003).
21. Guo Y.: Decay and Continuity of Boltzmann Equation in Bounded Domains. Arch. Rat. Mech. Anal. 197(3), 713-809 (2010).
22. Grad, H.: Asymptotic theory of the Boltzmann equation. II. Rarefied gas dynamics. In: Proceedings of the 3rd international Symposium, (Paris, 1962), Lawmann, J. A. (ed.), New York: Academic Press, 1963, pp. 26-59.
23. Hamdache K.: Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions. Arch. Rat. Mech. Anal. 119(4), 309-353 (1992).
24. Hörmander, L.: The analysis of linear partial differential operators. I-IV, Berlin: Springer-Verlag, 2005.
25. Hwang H.-J.: Regularity for the Vlasov-Poisson system in a convex domain. SIAM J. Math. Anal. 36(1), 121-171 (2004).
26. Hwang H.-J., Velazquez J.: Global existence for the Vlasov-Poisson system in bounded domains. Arch. Rat. Mech. Anal. 195(3), 763-796 (2010).
27. Kim, C.: Boltzmann equation with specular reflection in 2D domains, In preparation.
28. Lions, P.-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, III. J. Math. Kyoto Univ. 34, no. 2, 391-427, 429-461, 539-584, (1994).
29. Maslova, N.: Nonlinear evolution equations. Kinetic approach. Rivers Edge, NJ: World Scientific Publishing Co., 1993.
30. Maxwell, J.-C.: On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. Roy. Soc. London 170, Appendix 231-256 (1879).
31. Melrose R. B., Sjostrand J.: Singularities of boundary value problems I. Comm. Pure Appl. Math. 31(5), 593-617 (1978).
32. Mischler S.: On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system. Commun. Math. Phys. 210(2), 447-466 (2000).
33. Mouhot C., Villani C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rat. Mech. Anal. 173(2), 169-212 (2004).
34. Sone, Y.: Molecular gas dynamics. Theory, techniques, and applications. Modeling and Simulation in Science, Engineering and Technology. Boston, MA: Birkhauser Boston, Inc., 2007.
35. Sone Y., Takata S.: Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer. Transport Theor. Stat. Phys. 21, 501-530 (1992).
36. Takata S., Sone Y., Aoki K.: Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules. Phys. Fluids A 5, 716-737 (1993).
37. Taylor M.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28(4), 457-478 (1975).
38. Ukai, S.: Solutions of the Boltzmann equation. Patterns and waves. Stud. Math. Appl., 18, Amsterdam: North-Holland, 1986, pp. 37-96.
39. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics. Vol. I, Amsterdam: North-Holland, 2002, pp. 71-305.
40. Voigt, J.: Functional analytic treatment of the initial boundary value problem for collisionless gases. Habilitationsschrift, Munchen, 1981 (http://www. math. tu-dresden. de/~voigt/vopubl/habilschr/habil80. pdf).
41. Wennberg B.: Regularity in the Boltzmann Equation and the Radon Transform. Commun. in P. D. E. 19, 2057-2074 (1994).
42. Wennberg B.: The geometry of binary collisions and generalized Radon transforms. Arch. Rat. Mech. Anal. 139(3), 291-302 (1997).